What is the solution to this mathematical series of real numbers?

• Michael_0039
In summary, the conversation discusses finding the limit of a series and determining its convergence using various methods. The conclusion is that the series converges by comparing it with the well-known convergent series of the hyperharmonic series. Other methods such as the Cauchy condensation criterion or the integral test can also be used.
Michael_0039
Homework Statement
I used D'alembert criterion to determine if the mathematical sequence converges or diverges
Relevant Equations
Σ(1/(n+1)^2

You have a mistake in the last line, it is easy to prove that ##\lim_{n \to +\infty}(1-\frac{1}{n+2})^{n+2}=e^{-1}=\lim_{n \to +\infty}(1-\frac{1}{n+2})^{n}## so the final limit is 1.

Instead upper bound the series by the hyperharmonic series (p-series for p=2 )which is a well know result that it converges, hence by the comparison criterion this series converges too.

Michael_0039
Comparing with the convergent series

$$\sum_{n=1}^\infty \frac{1}{n^2} =\frac{\pi^2}{6}$$ yields the result.

Alternatively, use the Cauchy condensation criterium or the integral test.

Thanks for your answer. I wanted to use D'alembert criterion to see if a solution can be found.

1. What is the pattern or rule for this mathematical series?

The pattern or rule for a mathematical series is a set of instructions that can be used to determine the next number in the series. It could involve a specific mathematical operation, such as addition or multiplication, or it could be based on a specific sequence of numbers.

2. How can I solve this mathematical series?

To solve a mathematical series, you need to first identify the pattern or rule for the series. Then, you can use this pattern to determine the next number in the series. This process can be repeated until you reach the desired number in the series.

3. Can there be more than one solution to a mathematical series?

Yes, there can be more than one solution to a mathematical series. This is because there can be different patterns or rules that can be used to determine the next number in the series. However, there is usually one most commonly accepted solution for a particular series.

4. How can I check if my solution to a mathematical series is correct?

To check if your solution to a mathematical series is correct, you can use the same pattern or rule to determine the next few numbers in the series. If your solution matches these numbers, then it is likely that your solution is correct. You can also ask someone else to check your solution or use a calculator to verify your answer.

5. Are there any shortcuts or tricks for solving a mathematical series?

Yes, there are some shortcuts or tricks that can be used to solve certain types of mathematical series. For example, you can use algebraic equations or geometric formulas to quickly determine the next number in a series. However, these shortcuts may not work for all types of series, so it's important to also understand the underlying patterns and rules for solving a series.

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